MinT - Best Known (115−49, 115, s)-Nets in Base 9

Best Known (115−49, 115, s)-Nets in Base 9

(115−49, 115, 344)-Net over F₉ — Constructive and digital

Digital (66, 115, 344)-net over F₉, using

3 times m-reduction [i] based on digital (66, 118, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 59, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(115−49, 115, 452)-Net over F₉ — Digital

Digital (66, 115, 452)-net over F₉, using

1 times m-reduction [i] based on digital (66, 116, 452)-net over F₉, using
- trace code for nets [i] based on digital (8, 58, 226)-net over F₈₁, using
  - net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
      - a curve from the manYPoints database [i]

(115−49, 115, 41761)-Net in Base 9 — Upper bound on s

There is no (66, 115, 41762)-net in base 9, because

1 times m-reduction [i] would yield (66, 114, 41762)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 078870 604334 289218 862450 316162 216167 937885 847842 681883 184808 765275 212312 841685 010841 538163 018762 049498 716289 > 9¹¹⁴ [i]